y. f. yao and n. d. sandham- direct numerical simulation of turbulent trailing-edge flow with base...

832019 Y F Yao and N D Sandham- Direct Numerical Simulation of Turbulent Trailing-Edge Flow with Base Flow Control

httpslidepdfcomreaderfully-f-yao-and-n-d-sandham-direct-numerical-simulation-of-turbulent-trailing-edge 19

AIAA JOURNAL

Vol 40 No 9 September 2002

Direct Numerical Simulation of Turbulent Trailing-Edge Flowwith Base Flow Control

Y F Yaocurren and N D Sandhamdagger

Southampton University Southampton England SO17 1BJ United Kingdom

Direct numericalsimulationhas been carried out for turbulentow overa rectangular trailing edgeat a Reynolds

number of 1 poundpound 103 (based on the freestream quantities and the trailing-edge thickness) and ratio of boundary-

layer displacement thickness to trailing-edge thickness close to unity Two types of ow control were studied base

transpiration and secondary splitter plate Simulation of base transpiration was performed using different slit

heights and volume ow rates It was found that even small ow rates could produce signicant changes in overall

aerodynamic performance measured for example by the base pressure coefcient It was also found that for

the same volume ow rate a greater increase in base pressure (drag reduction) was obtained by blowing slowly

througha wide slitratherthan quickly througha narrow slit The effectiveness of a secondary splitter platelocated

on the trailing-edge centerline was investigated by varying the plate length from one to ve times the trailing-edge

thickness A signicant increase in the base pressure coefcient (about 25) was achieved even with the shortest

splitter plate equal to the trailing-edge thickness The b ase pressure coefcient increased monotonically with the

splitter plate length and no intermediate maximum value was found

Nomenclature

C p b = base pressure coefcienths = base slit height (width)k = turbulencekinetic energy L sp = length of secondary splitter platen = current time step p = instantaneouspressure pb = base pressure pref = reference pressureqb = base volume ow rate

Re h = Reynolds number based on freestreamvelocity and trailing-edge thickness Re plusmncurren = Reynolds number based on freestream

velocity and displacementthicknesst = time variableU = streamwise mean velocityub = base transpirationvelocityu i u v w = instantaneousvelocity variablesur = maximum reverse velocity at centrelineuiquest = friction velocity [du=d yw = Reh ]1=2

xi x y z = Cartesian coordinateaxes xr = recirculation length at centerline yC = y coordinate in wall units yu iquest Reh

1t = increment of time step

1 x

C

i = increments of coordinates xi in wall unitsplusmn99 = boundary-layerthicknessplusmncurren = boundary-layerdisplacement thicknessplusmncurrenin = boundary-layerdisplacement thickness at inlet

Introduction

T HEtrailing-edgeregionofairfoilsandturbinebladesinuencesthe entire aerodynamic performance measured for example

Received 23 August 2001 revision received 28 March 2002 acceptedfor publication 28 March 20 02 Copyright cdeg 2002 by Y F Yao and N DSandhamPublished by the American Instituteof Aeronauticsand Astronau-tics Inc with permission Copies of this paper may be made for personal

or internal use on condition that the copier pay the $1000 per-copy fee tothe Copyright Clearance Center Inc 222 Rosewood Drive Danvers MA01923 include the code 0001-145202 $1000 in correspondence with theCCCcurrenResearch Fellow Aeronautics and Astronautics School of Engineering

Sciences yyaosotonacukdagger Professor Aeronautics and Astronautics School of Engineering Sci-

ences nsandhamsotonacuk Senior Member AIAA

by the base pressure coefcient or total pressure loss Because theregion is highly localized it can be a good candidate for the intro-duction of ow control techniques aiming at drag reduction Nu-merically it is difcult to achieve an accurate simulation using theconventional Reynolds-averagedNavierndashStokes approach becauseof the complexity of the problem In the immediate vicinity of thetrailing edge the ow may be highly unsteady and far from anequilibriumstate and consequentlyconventionalturbulencemod-els either prove to be inadequate or require special modicationsHowever simplied versionsof the problemcan be computedusingdirect numerical simulation (DNS) Previous experience1 showedthat good quality results can be obtained at moderate Reynoldsnumber (raquo1pound103 basedon the trailing-edgethicknessor boundary-layer displacementthickness) with currently available computer re-sources The intention of this paper is therefore to study mecha-nisms of ow control in a model trailing-edge ow using DNS

Early researchwork on the applicationof owcontrol to the trail-ing edge was carried out by Nash2 Wood3 and Bearman4 aroundthe mid-1960swhen studies were made of the effects of base bleed(also called secondary airow) on the aerodynamicperformance of the two-dimensional blunt trailing edge in particular the increasein base pressure coefcient (and consequent decrease in drag co-efcient) Since then the base bleed concept has been applied toa wider range of practical problems for example in transonic andsupersonic ows5 In the gas-turbine industry a similar techniquecalledgas coolantejection has also been developed andused for thecontrolof turbine bladetemperatureto extendthe lifetimeof bladesExperiments haveshown that the method not only results in a lowerheat transfer to the blade but also leads to a signicant reductionin total pressure loss6 When the ejection velocity and angle weretuned an overall optimization could be achieved7 Although themethod clearly works it is still important to understand the under-lying ow mechanism On this point experimentalists have madesome progress and the mechanism of drag reduction is believedto involve a change in the wake dynamics8 Computational studieshaveprogressedwith improvements in computer power and numer-ical methods Hannemann and Oertel9 tackled the laminar wakeunsteadinessproblem and Hammond and Redekopp10 analyzedtheglobal instability of the ow These pilot investigationsshowed thatthe global dynamics associated with vortex sheddingcould be sup-pressedby base transpirationBased on the successof these laminarcalculations a fully turbulent ow study is desirable

An alternative way of achieving the base drag reduction is to usean additional (secondary) splitter plate along the wake centerlineAn important theoretical investigation of the effect of such a split-ter plate on the drag and pressure distribution of bluff body ow

1708



YAO AND SANDHAM 1709

was carried out by Roshko11 Later this simple but effective con-cept was applied in experiments1213 and a review paper was g ivenby Tanner14 With developments in experimental and computa-tional techniques there have been additional investigations in re-cent years1516 Compared to the blunt trailing-edge geometry thepresence of a splitter plate modies the near-wake vortex evolu-tion and thus limits the main vortex interactions in the recircula-tion zone It has b een conrmed both by experiments121315 and bytwo-dimensionalcomputation16 that the drag coefcient and vortex

shedding can be signicantly reduced by the use of a splitter plateof relatively short length

The present study is focused on the two types of ow control just describedThe computation includes a precursorsimulationof turbulent boundary-layerow and a successor simulation of turbu-lent trailing-edgeow Flow characteristicswith no ow control arestudied rst providing a benchmark for reference and illustratingthe main features of the vortex shedding and interaction mecha-nism Simulations with ow control using base transpiration and asecondary splitter plate follow with emphasis on the base pressurecoefcient ow structure and mean properties

Simulation

Governing Equations and Numerical Method

All variables dened in the Nomenclature are nondimensionalDistances are normalized with trailing-edge thickness velocitieswith freestreamvelocityand pressurewith freestreamdensitytimesthe square of freestream velocity

We consider an incompressible uid moving with velocityu i D u vw and pressure p in a Cartesian coordinate system xi D x y z and use t to denote the time The uid motion sat-ises the dimensionlessNavierndashStokes equationsg iven by

ui

t C

u j ui

x j

D iexcl p

xi

C1

Re h

2u i

x j x j

(1)

and the continuity equation

u i

xi

D 0 (2)

The Reynolds number Reh is based on the dimensional referencevelocity (chosen as the freestream velocity) and the dimensionalreferencelength(chosenas the trailing-edgethickness) which havebeen usedthroughoutthis paperto make all variablesdimensionless

The Navierndash Stokes equations are discretizedon a staggered gridusing a second-ordercentral nite differencescheme and advancedin time with the projectionmethod based on a second-orderexplicitAdamsndash Bashforthscheme The provisionalvelocity ucurren

i is projectedusing the data from the current time step n and preceding time stepn iexcl 1 as

ucurreniiexcl un

i

1t D

3

2 H n

iiexcl

1

2 H n iexcl 1

iC

1

2

pn iexcl 1

xi

(3)

where the quantity H i is dened as

H i D iexclu j ui

x j

C1

Reh

2u i

x j x j

This is then corrected for continuity to yield the velocity at a timestep nC 1 using

unC 1

iD ucurren

iiexcl

3

21t

pn

xi

(4)

where the pressure pn is obtained by the solution of the Poissonequation

2 pn

xi xi

D2

31t

ucurreni

xi

(5)

Turbulent Boundary-Layer Flow

Simulation of turbulentboundary-layerow was performedrstusing a spatial DNS code that implemented the rescaling and feed-back technique developed by Lund et al17 A computation box of size 50pound 20pound 6 (based on units of inlet displacement thicknessplusmncurrenin ) in the streamwise x wall-normal y and spanwise z directionswas used with 128pound 96pound 64 grid points stretched in the wall-normal direction and uniformly distributed in the streamwise andspanwisedirectionsA gridresolutionof 1 xC D 19 1 yC

1D 05and

1z

C

D 45 was achieved with about 10 points in the viscous sub-layer ( yC middot 10) The turbulentboundarylayer was simulatedwith aconditionof Replusmncurren D 950at theinow planeResults forskin-frictionand turbulence statistics were in good agreement with other DNSdata and experimental results1 A time sequence of ow variablesat a sampling plane near the exit was stored for later use as theinow boundary condition in the following turbulent trailing-edgesimulation

Turbulent Trailing-Edg e Flow

Simulations of turbulentow overa rectangulartrailing-edgege-ometry were carried out at a Reynolds number Reh D 1pound 103 Theupper and lower incoming turbulent boundary-layers were statisti-cally equivalent but uncorrelatedin time They developed along the

plate for a short distance beforecontactingeach other at the trailingedge The Cartesian coordinate system was built up with its originO at the center of the trailing edge

Computationswere carried out witha parallelcomplex-geometrysimulation code (see Thomas and Williams18 ) that has been par-allelized using the message passing interface (MPI) library andported to the Cray-T3E parallel computer Validations have beenmade for various cases such as backward-facing step ow open-channelow and skewed ow past a wall-mountedcube all givinggood agreement with published computational results and experi-mental data

The computational box size is 20pound 16pound 6 with the x length of 20 split into 5 along the at plate and 15 in the wake region in thestreamwise x direction The upstreamdistance of 5 is chosen based

on the experiment observation by Gough and Hancock

19

that theboundarylayer morethan plusmn99 aheadof thetrailingedgeis unaffectedby the presence of the trailing edge as well as the numerical studyby Hannemann and Oertel9 that the rms uctuationsat this locationwere more than two orders of magnitude lower than the maximumrms uctuations occurring in the wake Lengths of 85 on the uppersideand75onthelowersideinthewall-normal y directionare usedwhereas the length in the spanwise z direction is 6 (Fig 1) Thisasymmetric arrangement in y was chosen mainly on the groundsof load balancing on the parallel computer used in this study butalso as a check on the box size because any asymmetries in theresults would imply that the computational box was too small inthe y direction Grid points are uniformly distributed in all threedirectionswith 256pound 512pound 64 points in x y and z r espectivelyA

detailed study of domain size dependency and grid resolution wascarried out by Yao et al1 with the conclusionthat this congurationand grid resolution are suitable for the current simulations

The boundary conditions are dened as follows At the inletthe data are prescribed and taken from the precursor turbulent

Fig 1 Computational box for the trailing-edge simulation with and

without ow control (not to scale)



1710 YAO AND SANDHAM

Fig 2 Streamlines in the near-wake region with no ow control

Top view Side view

Fig 3 Instantaneous ow structures in the near-wake region from the simulation with no ow control spanwise vortices appear in the darker shade

showing a pressure isosurface with p =iexcl

0

035 whereas streamwise vortices in the lighter shade showP

vortices withP

= 0

4

boundary-layer simulation The outlet plane is treated with a stan-dard convective condition so that outgoing disturbances can leavethe domain smoothly On the top and bottom boundaries of the do-main free-slip conditions are used and in the spanwise directiona periodic condition is applied A no-slip condition is used on allsolid walls including the splitter plate when present

Results and Discussion

Reference Case No Flow Control

For referencewe rst give a brief overviewof results for the casewith no ow control More complete sets of results for this case

were reported by Yao et al1

and Thomas et al20

Figure 2 showsthe streamline plots in the near wake (Only one half-domain isplotteddue to thesymmetry) A recirculationregion witha lengthof xr rsquo 20 (where xr is denedas the dimensionlessdistancebetweenthe trailing-edge end plane and the stagnation point in the wake)exists after the trailing edge The maximum reverse velocity ur isabout 10 of the freestreamvelocity The turbulencekineticenergyreaches its peak value of about 003 at the centerline in the nearwake and then decaysdownstream The base pressure coefcient isdened by

C p b D 2 pb iexcl pref (6)

where the averaged dimensionlessbase pressure pb is calculatedbyintegrating the static pressure over the trailing-edgee nd plane andthe reference dimensionless pressure pref is taken as the freestreamvalueat the inletplane The simulation(which itselfuses a pressurerelative to a xed point on the solid wall upstream of the trailingedge) gives pb Diexcl00695 and pref Diexcl0036 The base pressurecoefcient is thereforeiexclC p b D 00669

Snapshots of an instantaneous oweld are shown in Fig 3in a subdomain of 10pound 6pound 6 This illustrates the interaction be-

a) b)

Fig 4 Congurationsof the two types of ow control a) base transpi-ration and b) secondary splitter plate the trailing-edge thickness is set

to be 1

tween the large-scale spanwise vortices (presented in the darkershadean isosurfaceof pressurecontaininga low-pressurecore) andsmaller-scale streamwise vortices (presented in the lighter shadean isosurface of the second invariant of the velocity gradient ten-sor [5D u i = x j u j = xi )] named 5 vortices for later use) Itcan be seen that the well-known von Karman vortex street existsin the near wake The appearanceof vortex shedding is related (viaa global mode10 ) to the existence of absolute instability based onlinear stability theory (see Hannemann and Oertel19 )

Flow Control by Base Transpiration

Simulations with base transpiration were carried out using thearrangement shown in Fig 4a A total of 10 simulations (cases 1ndash

10) were carried out with different slit heights (hs D 025 05 and10) and volume ow rates per unit span (qb Dsect0025sect005 andsect01) dened as qb D ub hs where ub is the uniform base owvelocity The particular combinations chosen for hs ub and thecorresponding qb are shown in Table 1

The overall effectivenessof the ow control can be measured bythe effect on the base pressure coefcient and the base pressure




These are plotted in Fig 5 The near collapse of the curves whenplotted against volume ow rate in this manner illustrates that vol-ume ow rate is the key parameter Base pressure drops sharplywhen suction is applied The corresponding base pressure coef-cient decreases (and consequentlydrag increases) by up to a factorof four for suction with volume ow rate qb Diexcl01 For base blow-ing the opposite effect is observed with a maximum increase of base pressure coefcient (drag reduction) of approximately a fac-tor of two for qb D 01 However for blowing the effectiveness is

strongly affected by slit width hs and it is clear that for the samevolume ow rate a greater increase in base pressure coefcient isobtained by blowing slowly through a wide slit rather than quicklythrough a narrow slit This is in good agreement with the exper-imental observation that the best drag reduction for a given massow can be obtained by ejecting the bleed air at the lowest possiblevelocity314

The effect of base transpiration on the mean ow is revealed bystreamlineplots Figure 6 shows the effect of varyingthe base tran-spiration through a slit height of hs D 025 Three cases are shownwith ub D iexcl01 01and 04and consequentlyqb equaltoiexcl00250025 and 01 For weak suction ow qb Diexcl0025 we see that therecirculation length xr has reduced to about 158 compared to 20for the reference case of no ow control The closed recirculation

zoneis locatedbetween theedgeof theslit yD 00125andtheedgeof the plate yD 05 For weak blowing ow qb D 0025 the mainrecirculationhas moved downstream starting at x D 10 and endingat x D 248 giving a total recirculation length of xr D 148 A sec-ondary recirculation is also observed just above the slit adjacent

Table 1 Simulationswith two types of ow control base

transpiration and a secondary splitter plate

Case h s ub qb xr ur Lsp

Ref 00 00 00 20 iexcl01 01 025 iexcl04 iexcl01 105 iexcl04 mdashmdash2 025 iexcl01 iexcl0025 158 iexcl0124 mdashmdash3 025 C01 C0025 148 iexcl0046 mdashmdash4 025 C04 C01 mdashmdash mdashmdash mdashmdash5 05 iexcl02 iexcl01 098 iexcl025 mdashmdash6 05 iexcl01 iexcl005 145 iexcl0154 mdashmdash7 05 C01 C005 mdashmdash mdashmdash mdashmdash8 05 C02 C01 mdashmdash mdashmdash mdashmdash9 10 iexcl01 iexcl01 10 iexcl02 mdashmdash10 10 C01 C01 mdashmdash mdashmdash mdashmdash11 mdashmdash mdashmdash 00 300 iexcl00849 112 mdashmdash mdashmdash 00 334 iexcl00805 213 mdashmdash mdashmdash 00 363 iexcl0045 314 mdashmdash mdashmdash 00 391 mdashmdash 415 mdashmdash mdashmdash 00 421 mdashmdash 5

Fig 5 Base pressure coefcient (left side solid symbols) and base pressure (right-hand side open symbols) as a function of volume ow rate

a)

b)

c)

Fig 6 Streamlines for base transpiration through a slit of height

h s = 025 with volume ow rates of q b a ) iexcl 0025 b) 0025and c) 01




to the trailing edge For very strong blowing ow qb D 01 there isno downstream recirculation zone but now two small recirculationzones are seen in the very near wake region between the slit andthe edge of the plate Streamwise mean velocity U and turbulencekinetic energy k variations along the centerline ( y D 0) are shownin Figs 7 and 8 for all of the cases with hs D 025 and hs D 05respectively It can be seen that in general as qb is decreased therecirculation zone (negative U ) moves closer to the trailing edgeand themagnitudeof the (negative) peakincreasesaccompaniedby

an increase in k From Figs 7 and 8 it appears that except for thecase with the strongestsuction the peak value of k occurs approx-imately twice as far from the trailing edge as the local minimumin U As a summary values of the maximum recirculationvelocityur and the recirculation length xr for all cases run are shown inTable 1

Instantaneous ow structures show the effect of base transpi-ration on the structure of the near wake Figure 9 shows snap-shots from the cases of strong suction (hs D 10 ub Diexcl01 andqb Diexcl01) and strong blowing (hs D 10 ub D 01 and qb D 01)illustrating the ow structure with isosurfaces of pressure and 5as shown earlier for the reference case in Fig 3 For the strongsuction case it can be seen that the quasi-two-dimensionalcoher-ent structure has been strengthened considerably The strong span-

wise vortex structures even appear upstream of the trailing edgeSubsequently the 5 vortices also become stronger because of theinteractions Conversely with strong blowing the structures arepushedfartherdownstream and become much weaker As a result asmaller peak value of k is evident (Figs 7 and 8) There also seemsto be tendency to lose the strong spanwise coherence of the pres-sure structures for the case with strong blowing and the sheddinghas been moved farther downstreamagreeingqualitativelywith theexperiment4

Suction or blowing also affects the skin friction on the plate up-stream of the trailing edge with suctionincreasingthe friction dragand blowing decreasing it However this effect was at most only97 of the base pressure drag change

Fig 7 Mean velocity U and turbulence kinetic energy k distributionsfor slit height h s = 025 the reference simulation of no ow control is shown bythe solid line

Flow Control by Secondary Splitter Plate

An alternativetechnique of ow control is the addition of a zero-thickness secondary splitter plate behind the trailing edge as illus-trated in Fig 4b In this study the splitter plate length L sp is variedover a range of 1ndash 5 and results are compared with those from thereference case of no ow control and experimentaldata

Figure 10 gives the base pressure coefcient and base pressurevariationsas a function of splitter plate length L sp With Lsp D 5 amaximum reductionof iexclC p b of about44 is achievedHowever

a splitter plate with L sp D 1 already gives a 25 reductionShorterplatesare probablyalso preferredgiven practical(structural) consid-erations A maximum drag reduction of 50 was achieved experi-mentally by Bearman13 fora owwithMach numberapproximately01 and splitter plate length L sp D 4 However compared to t he re-sults of Nash et al12 who studied the secondary splitter plate for arectangular trailing-edge geometry at Mach number 04 there arelarge differences The magnitudes of the reductions of iexclC p b arehere much smaller than those in the Nash et al experiment whichfound a factor of two reduction for L sp D 1 and a factor of threereduction for Lsp D 4 One possible cause for the difference is thereferencepressureheretaken as the freestreamat the computationalinow upstreamof the trailingedge whereas in the experimentsit ispresumablytakenfartherawayfrom the objectOtherdifferencesare

the Mach number (incompressibleow was assumed in the currentcomputation) and Reynolds number (lower in the computations)

A phenomenon of a local maximum in the base pressure coef-cientasafunctionof L sp was observedexperimentallyby Bearman13

at Reynolds numbers between 14pound 105 and 256pound 105 in a low-speed wind tunnel A peak appeared at L sp D 15 and increasedsignicantly with the decrease of Reynolds number indicating astrong Reynolds number effect Such a peak was also observed inthe experiments of Nash et al12 However the phenomenon was notfoundin a recentexperimentby Rathakrishnan15 usinga rectangularcylinder in a low-speedwind tunnel (Reynolds numbers 058pound 105

and 098pound 105 ) The experiments showed a signicant decrease iniexclC p b for Lsp D 1 (about a 40 reduction) but the effect of Lsp




Fig 8 Mean velocity U and turbulence kinetic energy k distributionsfor slit height h s = 05 the reference simulationof no ow control is shown with

the solid line

a) b)

Fig 9 Instantaneous ow structures in the near-wake region (shade denition as in Fig 3) a) side view with ba se suction h s =10 u b =iexcl 01 andq b = iexcl 01 and b) top view with base blowing h s =10 u b =01 and q b =01

Fig 10 Base pressure coefcient (left-hand side square symbols) and base pressure (right-handside triangle symbols) as a functionof the secondarysplitter plate length




beyond that was found to be insignicant A satisfactory physicalexplanation for the appearance of a local drag minimum has notbeen offered to date Experiments and simulations using a widerrange of Reynolds number are needed

Figure 11 shows streamline plots with splitter plate lengths of L sp D 1 and 5 The size of the recirculation grows as the splitterplate length increases although the bubble shape remains broadlyunchanged A summary of the ve cases (cases 11ndash 15) is given inTable 1 Compared to the referencecase of no ow controlthe sep-

aration length xr increases and the maximum reverse velocity ur

decreaseswith splitterplate length The recirculationterminatesaf-ter the end ofthe secondaryplate for L sp D 1 2and 3 which agreeswith the experimental observation by Bearman13 who claimed thatno reattachment was found for shorter splitter plates Lsp D 1raquo 2However for the longest two plates the simulated recirculationzone terminates on the plate that is the ow behaves broadly asa backward-facing step ow These two cases have different reat-tachment lengths indicatingthat the ow upstream of reattachmentis affectedby the ow developmentsdownstreamThe recirculationlengths from the simulations ( xr D 391 for the L sp D 4 case and

Lsp = 1 Lsp = 5

Fig 11 Streamlines in the near wake with a secondary splitter plate

Fig 12 Mean velocity U and turbulence kinetic energy k distributions of Lsp = 1iexcl 5 compared with the reference simulation

xr D 421 for the L sp D 5 case) are longer than experiment13 wherea recirculationlength of xr D 29 was observed for L sp D 4 DNS of a backward-facing step ow21 gave a mean value of 628 (based onthe step height that is half of the trailing-edge thickness) equiv-alent to 314 based on the trailing-edge thickness However notethat the Reynolds number used in the current simulation (5pound 102

based on the equivalent step height) is much lower than that in theexperiment and 10 times lower than that used in the Le et al 21

DNS (51pound 103 based on the step height) Also the ratio of the

boundary-layerthickness to the step height is close to unity in thestudy of Le et al21 whereas in the present case the boundary-layerthickness is about 13 times the half-thickness of the trailing edgeAnother signicant difference to the backward-facing step ow isthat no secondaryseparationwas observedin the simulationswhichmay also be a low Reynoldsnumber effect The friction drag on thesecondary plate was found to be small (6 at most) compared tothe base pressure drag

Figure 12 gives a comparison of the mean velocity and the tur-bulence kinetic energy distributions along the centerlineIt is clearthat even a short splitter plate with a lengthof L sp D 1 that is equal




a)

b)

c)

Fig 13 Turbulence kinetic energy budget in the middle of the recir-culation regiona ) no ow control x=10b) base transpiration x =17with h s =025 u b = 01 and q b = 0025 and c) secondary splitter plate

x = 18 with Lsp = 1 The thin solid line represents the production termand the thick solid line represents the balance (sum of a ll terms)

to the trailing-edge thickness has a signicant effect on the meanvelocity and the turbulence kinetic energy distributions

Energy BudgetsEnergy budgets havebeen analyzed usingstatisticaldata from the

simulationsThe turbulencekinetic energy equation is

k

t C R D P iexcl sup2 iexcl

iexcl J u

iC J

p

iC J ordm

i

cent

xi

(7)

where R D hu iik = xi ) is the convection term P is the produc-tion term sup2 is the dissipation and J u

i J

p

i and J ordm

iare turbulence

a)

b)

c)

Fig14 Turbulencekineticenergy budget atthe endof therecirculationregiona ) no owcontrol x=20b) base transpiration x= 25 withh s =025u b = 01andq b = 0025andc) secondary splitter plate x = 29with

Lsp = 1 The thinsolid linerepresents the productionterm and thethicksolid line represents the balance (sum of all terms)

triple-moment(transport) pressure-diffusionand viscous transportterms respectivelydened in detail by Yao et al1

Turbulence kinetic energy budgets were computed for three dif-

ferent cases no ow control ow control with base transpiration(hs D 025 ub D 01 and qb D 0025) and ow control with sec-ondarysplitter plate ( L sp D 1) The comparisons were carriedout attwo selected positions the middle of the recirculation region andthe end of the recirculation region Results are shown in Figs 13and 14

At the middle of the recirculation region for the reference caseof no ow control(Fig 13a) the convection term is very signicant




with a positivepeakat y rsquo 05 anda negativepeak (correspondingtoa reverseow) at the centerline y D 0 Theproductionterm also hasa small negative peak near the centerline and a high positive peak away fromit Thenegativeproductionwasstudiedby Thomas et al20

A strongpressure-diffusionterm with the same order of magnitudeas the productionand convection terms is also observed It impliesthat the pressure-diffusioneffect cannotbe neglected in the trailing-edge ow The dissipation has a maximum at the same position asthe maximum productionbut is onlyapproximately30 of thepeak

productionvalue Hencethe ow is locallyfar from an equilibriumstate When base transpirationis applied (Fig 13b) the productionand dissipation terms remain broadly unchanged whereas the con-vection and pressure-diffusion terms are both reduced Howevera signicant negative peak in turbulence triple-moment (transport)term appears again at the same peak position of the production anddissipationterms With the secondarysplitter plate (Fig 13c) a sig-nicant decrease in the peak production (almost half of the valuecompared to that in the referencecase) is evidentwhereas the otherterms are with the exception of the triple-moment term similar tothose found in the base transpiration case

At the end of the recirculation region for the case of no owcontrol (Fig 14a) the production term dominates the energy bud-get with signicant contributions from the convection turbulence

triple-momentpressure-diffusionand dissipationterms The pres-sure diffusionchanges from positive to negativeas the centerline isapproachedFor the two types of ow controlconsidered (Figs 14band 14c) the peak value of all terms decreases with the pressurediffusionbeing moresignicantlyaffectedthan the othertermsThetriple-moment term is relatively less important for the base transpi-ration (blowing) case than for either the reference case or for thesecondary splitter plate case L sp D 1

The detailedbehaviorof terms in the energy budget discussedear-lier is important for constructing models at the two-equation levelwhich can subsequentlybe used for higher Reynolds number stud-ies and optimization of control schemes However the main resultscan be summarized more simply At the end of the recirculationzone (Figs 14b and 14c) we are seeing the result of ow control

in that all turbulence activity is diminished whereas in the middleof the recirculation we see differences in detail between the twocontrol schemes In particular the secondary splitter plate methodis more effective at reducing turbulenceactivity within the recircu-lation zone

Conclusions

It can be concluded from this study that DNS is feasible for theparametricstudy of ow controlin the trailing-edgeow usingbasetranspiration and a secondary splitter plate Both methods of owcontrol have a signicant effect on the aerodynamic performancemeasured for example by the base pressure coefcient Simula-tions with base transpiration have been carried out with differentslit heights and volume ow rates It was revealed from the simula-

tionthatsuction(even weak suction) enhances the near-wakevortexshedding sustains the coherent structure over an extended regionand shortensthe recirculationregionwhereasstrongblowingmovesthe entireinteractiondownstreamand diminishesthe coherentstruc-ture more quickly It was also found that for the same volume owrate a greater base pressure increase can be obtained by blowingslowly througha wide slit ratherthan quickly througha narrow slitin agreementwith experiments Simulations with secondarysplitterplate showed a monotonicincreasein base pressure coefcient withthe splitter plate lengthA 25 increase was achieved for the split-ter plate equal to the trailing-edge thickness whereas a maximum44 increase was achieved for the splitter plate length ve timesthe trailing-edge thickness Note that in practical applications thebase pressure drag is not the only consideration and the drag re-

ductionproblem must be considered in the context of aerodynamicand structural constraintsThe turbulence kinetic energy budget inthe recirculation region shows that the pressure-diffusion term hasa signicant contribution to the budget balance whereas the peak

of dissipationis only about 30 of the peak productionThe effectof ow control is in general to reduce the convectionand pressuretransport relative to the production in the core of the recirculationAt the end of the recirculation region the structure of the budget isretained but the magnitudes of all terms are reduced

Acknowledgments

The authorswouldliketo thankthe UnitedKingdomEngineeringand Physical Science Research Council for nancial and parallel

computer support through Grants GRL 18570 and GRM 08424

References1 Yao Y F Thomas T G Sandham N D and Williams J J R ldquoDi-

rect Numerical Simulation of Turbulent Flow over a Rectangular TrailingEdgerdquo Theoreticaland ComputationalFluidDynamicsVol14No52001pp 323ndash336

2 Nash J F ldquoA Discussion of Two-Dimensional Turbulent Base FlowsrdquoAeronautical Research Council TR RampM 3468 London 1967

3 Wood C J ldquoThe Effect of Base Bleed on a Period ic Wakerdquo Journal of

the Royal Aeronautical Society Vol 68 July 1964 pp 477ndash4824 Bearman P W ldquoThe Effect of Base Bleed on the Flow Behind a Two-

Dimensional Model with a Blunt Trailing Edgerdquo Aeronautical QuarterlyVol 18 Aug 1967 pp 207ndash224

5 Motallebi F and Norbury J F ldquoEffect of Base Bleed on Vortex Shed-

ding and Base Pressure in Compressible Flowrdquo Journalof FluidMechanicsVol 110 1981 pp 273ndash 2926 Deckers M and Denton J D ldquoAerodynamicsof Trailing-Edge-Cooled

Transonic Turbine Blades Part 1mdashExperimental Approachrdquo American So-ciety of Mechanical Engineers ASME Paper 97-GT-518 1997

7 Pappu K R and Schobeiri M T ldquoOptimization of Trailing EdgeEjection Mixing Losses A Theoretical and Experimental Studyrdquo AmericanSociety of Mechanical Engineers ASME Paper 97-GT-523 1997

8 Bearman P W ldquoNear Wake FlowsBehind Two- and Three-DimensionalBluff Bodiesrdquo Journal of Wind Engineering and Industrial AerodynamicsVol 49 No 71 1997 pp 33ndash 54

9 HannemannK andOertel H ldquoNumerical Simulation of the Absolutelyand Convectively Unstable Wakerdquo Journal of Fluid Mechanics Vol 1991989 pp 55ndash 88

10 Hammond D A and Redekopp L G ldquoGlobal Dynamics and Aero-dynamic F low Vectoring of Wakesrdquo Journal of Fluid Mechanics Vol 338

1997 pp 231ndash24811 Roshko A ldquoOn the Wake and Drag of Blunt Bodiesrdquo Journal of

Aerospace Sciences Vol 22 Feb 1955 pp 124ndash13212 Nash J F Quincey V G and Callinan J ldquoExperiments on Two-

Dimensional Base Flow at Subsonic and Transonic Speedsrdquo AeronauticalResearch Council TR RampM 3427 London 1963

13 Bearman P W ldquoInvestigation of the Flow Behind a Two-DimensionalModel with a Blunt Trailing Edge and Fitted with Splitter Platesrdquo Journal

of Fluid Mechanics Vol 21 19 65 pp 241ndash 25514 Tanner M ldquoReduction of Base Dragrdquo Progress in Aerospace Sciences

Vol 16 No 4 1975 pp 369ndash 38415 Rathakrishnan E ldquoEffect of Splitter Plate on Bluff Body Dragrdquo AIAA

Journal Vol 37 No 9 1999 pp 1125 112616 Park W C and Higuchi H ldquoNumerical Investigation of Wake Flow

Control by a Splitter Platerdquo KSME International Journal Vol 12 No 1

1998 pp 123ndash

13117 LundTS Wu Xand SquiresK DldquoGeneration of TurbulentInowConditions for Boundary Layer Simulationsrdquo Journal of Computational

Physics Vol 140 No 2 1998 pp 233ndash25818 Thomas T G and Williams J J R ldquoDevelopment of a Parallel Code

to Simulation Skewed Flowover a Bluff Bodyrdquo Journalof WindEngineering

and In dustrial Engineering Vol 67ndash68 Aprilndash June 1997 pp 155ndash16719 Gough T D and Hancock P E ldquoLower Reynolds Number Turbu-

lent Near Wakesrdquo Advances in Turbulence VI edited by S Gavrilakis LMachiels and P A Monkewitz Kluwer Academic Norwell MA 1996pp 445ndash448

20 Thomas T G Yao Y F and SandhamN D ldquoStructureand Energet-ics of a Turbulent Trailing Edge Flowrdquo Computers and Mathematics with Applications(to be published)

21 LeH MoinPand Kim JldquoDirectNumerical Simulationof TurbulentFlow over a Backward-Facing Steprdquo Journal of Fluid Mechanics Vol 330

1997 pp 349ndash374

W J Devenport Associate Editor




was carried out by Roshko11 Later this simple but effective con-cept was applied in experiments1213 and a review paper was g ivenby Tanner14 With developments in experimental and computa-tional techniques there have been additional investigations in re-cent years1516 Compared to the blunt trailing-edge geometry thepresence of a splitter plate modies the near-wake vortex evolu-tion and thus limits the main vortex interactions in the recircula-tion zone It has b een conrmed both by experiments121315 and bytwo-dimensionalcomputation16 that the drag coefcient and vortex

shedding can be signicantly reduced by the use of a splitter plateof relatively short length

The present study is focused on the two types of ow control just describedThe computation includes a precursorsimulationof turbulent boundary-layerow and a successor simulation of turbu-lent trailing-edgeow Flow characteristicswith no ow control arestudied rst providing a benchmark for reference and illustratingthe main features of the vortex shedding and interaction mecha-nism Simulations with ow control using base transpiration and asecondary splitter plate follow with emphasis on the base pressurecoefcient ow structure and mean properties

Simulation

Governing Equations and Numerical Method

All variables dened in the Nomenclature are nondimensionalDistances are normalized with trailing-edge thickness velocitieswith freestreamvelocityand pressurewith freestreamdensitytimesthe square of freestream velocity

We consider an incompressible uid moving with velocityu i D u vw and pressure p in a Cartesian coordinate system xi D x y z and use t to denote the time The uid motion sat-ises the dimensionlessNavierndashStokes equationsg iven by

ui

t C

u j ui

x j

D iexcl p

xi

C1

Re h

2u i

x j x j

(1)

and the continuity equation

u i

xi

D 0 (2)

The Reynolds number Reh is based on the dimensional referencevelocity (chosen as the freestream velocity) and the dimensionalreferencelength(chosenas the trailing-edgethickness) which havebeen usedthroughoutthis paperto make all variablesdimensionless

The Navierndash Stokes equations are discretizedon a staggered gridusing a second-ordercentral nite differencescheme and advancedin time with the projectionmethod based on a second-orderexplicitAdamsndash Bashforthscheme The provisionalvelocity ucurren

i is projectedusing the data from the current time step n and preceding time stepn iexcl 1 as

ucurreniiexcl un

i

1t D

3

2 H n

iiexcl

1

2 H n iexcl 1

iC

1

2

pn iexcl 1

xi

(3)

where the quantity H i is dened as

H i D iexclu j ui

x j

C1

Reh

2u i

x j x j

This is then corrected for continuity to yield the velocity at a timestep nC 1 using

unC 1

iD ucurren

iiexcl

3

21t

pn

xi

(4)

where the pressure pn is obtained by the solution of the Poissonequation

2 pn

xi xi

D2

31t

ucurreni

xi

(5)

Turbulent Boundary-Layer Flow

Simulation of turbulentboundary-layerow was performedrstusing a spatial DNS code that implemented the rescaling and feed-back technique developed by Lund et al17 A computation box of size 50pound 20pound 6 (based on units of inlet displacement thicknessplusmncurrenin ) in the streamwise x wall-normal y and spanwise z directionswas used with 128pound 96pound 64 grid points stretched in the wall-normal direction and uniformly distributed in the streamwise andspanwisedirectionsA gridresolutionof 1 xC D 19 1 yC

1D 05and

1z

C

D 45 was achieved with about 10 points in the viscous sub-layer ( yC middot 10) The turbulentboundarylayer was simulatedwith aconditionof Replusmncurren D 950at theinow planeResults forskin-frictionand turbulence statistics were in good agreement with other DNSdata and experimental results1 A time sequence of ow variablesat a sampling plane near the exit was stored for later use as theinow boundary condition in the following turbulent trailing-edgesimulation

Turbulent Trailing-Edg e Flow

Simulations of turbulentow overa rectangulartrailing-edgege-ometry were carried out at a Reynolds number Reh D 1pound 103 Theupper and lower incoming turbulent boundary-layers were statisti-cally equivalent but uncorrelatedin time They developed along the

plate for a short distance beforecontactingeach other at the trailingedge The Cartesian coordinate system was built up with its originO at the center of the trailing edge

Computationswere carried out witha parallelcomplex-geometrysimulation code (see Thomas and Williams18 ) that has been par-allelized using the message passing interface (MPI) library andported to the Cray-T3E parallel computer Validations have beenmade for various cases such as backward-facing step ow open-channelow and skewed ow past a wall-mountedcube all givinggood agreement with published computational results and experi-mental data

The computational box size is 20pound 16pound 6 with the x length of 20 split into 5 along the at plate and 15 in the wake region in thestreamwise x direction The upstreamdistance of 5 is chosen based

on the experiment observation by Gough and Hancock

19

that theboundarylayer morethan plusmn99 aheadof thetrailingedgeis unaffectedby the presence of the trailing edge as well as the numerical studyby Hannemann and Oertel9 that the rms uctuationsat this locationwere more than two orders of magnitude lower than the maximumrms uctuations occurring in the wake Lengths of 85 on the uppersideand75onthelowersideinthewall-normal y directionare usedwhereas the length in the spanwise z direction is 6 (Fig 1) Thisasymmetric arrangement in y was chosen mainly on the groundsof load balancing on the parallel computer used in this study butalso as a check on the box size because any asymmetries in theresults would imply that the computational box was too small inthe y direction Grid points are uniformly distributed in all threedirectionswith 256pound 512pound 64 points in x y and z r espectivelyA

detailed study of domain size dependency and grid resolution wascarried out by Yao et al1 with the conclusionthat this congurationand grid resolution are suitable for the current simulations

The boundary conditions are dened as follows At the inletthe data are prescribed and taken from the precursor turbulent

Fig 1 Computational box for the trailing-edge simulation with and

without ow control (not to scale)





Top view Side view



0


vortices withP

= 0

4






and Thomas et al20





a) b)


to be 1


















a)

b)

c)
























the solid line

a) b)










Lsp = 1 Lsp = 5











a)

b)

c)






k


iexcl J u

iC J

p

iC J ordm

i

cent

xi

(7)


i J

p

i and J ordm

iare turbulence

a)

b)

c)

















Conclusions





Acknowledgments
























1998 pp 123ndash








1997 pp 349ndash374






Top view Side view



0


vortices withP

= 0

4






and Thomas et al20





a) b)


to be 1


















a)

b)

c)
























the solid line

a) b)










Lsp = 1 Lsp = 5











a)

b)

c)






k


iexcl J u

iC J

p

iC J ordm

i

cent

xi

(7)


i J

p

i and J ordm

iare turbulence

a)

b)

c)

















Conclusions





Acknowledgments
























1998 pp 123ndash








1997 pp 349ndash374














a)

b)

c)
























the solid line

a) b)










Lsp = 1 Lsp = 5











a)

b)

c)






k


iexcl J u

iC J

p

iC J ordm

i

cent

xi

(7)


i J

p

i and J ordm

iare turbulence

a)

b)

c)

















Conclusions





Acknowledgments
























1998 pp 123ndash








1997 pp 349ndash374























the solid line

a) b)










Lsp = 1 Lsp = 5











a)

b)

c)






k


iexcl J u

iC J

p

iC J ordm

i

cent

xi

(7)


i J

p

i and J ordm

iare turbulence

a)

b)

c)

















Conclusions





Acknowledgments
























1998 pp 123ndash








1997 pp 349ndash374









Lsp = 1 Lsp = 5











a)

b)

c)






k


iexcl J u

iC J

p

iC J ordm

i

cent

xi

(7)


i J

p

i and J ordm

iare turbulence

a)

b)

c)

















Conclusions





Acknowledgments
























1998 pp 123ndash








1997 pp 349ndash374





a)

b)

c)






k


iexcl J u

iC J

p

iC J ordm

i

cent

xi

(7)


i J

p

i and J ordm

iare turbulence

a)

b)

c)

















Conclusions





Acknowledgments
























1998 pp 123ndash








1997 pp 349ndash374












Conclusions





Acknowledgments
























1998 pp 123ndash








1997 pp 349ndash374


y. f. yao and n. d. sandham- direct numerical simulation of turbulent trailing-edge flow with base...

Documents